A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 8, 9, 8, 6, 21, 14, 20, 26, 31, 24, 53, 35, 60, 68, 78, 76, 140, 115, 163, 183, 232, 218, 343, 301, 433, 432, 565, 542, 774

Offset

0,13

Comments

A partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are distinct, so y is counted under A239455(9).

A partition is Wilf iff it has distinct multiplicities of parts. For example, (2,2,2,1,1,1) has multiplicities (3,3), so is not counted under A098859(9).

The Heinz numbers of these partitions are given by A351294 \ A130091.

Is a(17) = 0 the last zero of the sequence?

Links

Table of n, a(n) for n=0..48.

Formula

a(n) = A239455(n) - A098859(n). Here we assume A239455(0) = 1.

Example

The a(9) = 1 through a(18) = 5 partitions are (empty columns not shown):
  n=9:      n=12:       n=15:         n=16:       n=18:
  --------------------------------------------------------------
  (222111)  (333111)    (333222)      (33331111)  (444222)
            (22221111)  (444111)                  (555111)
                        (2222211111)              (3322221111)
                                                  (32222211111)
                                                  (222222111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&Select[Permutations[#], UnsameQ@@Length/@Split[#]&]!={}&]], {n, 0, 15}]

Crossrefs

Wilf partitions are counted by A098859, ranked by A130091.

Look-and-Say partitions are counted by A239455, ranked by A351294.

Non-Wilf partitions are counted by A336866, ranked by A130092.

Non-Look-and-Say partitions are counted by A351293, ranked by A351295.

A000569 = number of graphical partitions, complement A339617.

A032020 = number of binary expansions with all distinct run-lengths.

A044813 = numbers whose binary expansion has all distinct run-lengths.

A225485/A325280 = frequency depth, ranked by A182850/A323014.

A329738 = compositions with all equal run-lengths.

A329739 = compositions with all distinct run-lengths

A351013 = compositions with all distinct runs.

A351017 = binary words with all distinct run-lengths, for all runs A351016.

A351292 = patterns with all distinct run-lengths, for all runs A351200.

Cf. A000041, A008284, A018783, A047966, A181819, A182857, A238130, A305563, A319149, A351203, A351204, A351290.

Sequence in context: A263414 A162934 A303908 * A082660 A134673 A131636

Adjacent sequences: A351589 A351590 A351591 * A351593 A351594 A351595

Keyword

nonn,more

Author

Gus Wiseman, Feb 16 2022

Status

approved